Touring 19th May till 24th May

Around Pentecost and the week after you can expect high booking ratios.
Even if not booked out you will likely save money by early booking.

Other renowned rental car firms are Sixt, Europcar and Avis. Personally I prefer Sixt but imo more a taste thing.

I would book 2-3 REFUNDABLE nights in one Mosel base town and the remaining nights in one Rhine base town. The areas are compact enough that you can head out and return to your base later the same day without a strict plan of attack and without excessive driving time. Pre-booking means getting the room/apartment and location that best suits you and your wallet. Booking on the fly could mean overpriced and undesirable leftovers... and some wasted time hunting those places down.

While I've rented cars for the area in the past, I prefer the train system for getting around both river valleys, which are well known as Germany's most scenic train rides. Rails connect almost all the towns that matter, cruise boats or shuttle buses serve the others, and cheap day passes allow for spontaneous travel decisions and hop-on, hop-off travel... no reservations needed.

Mosel River Scene: http://www.zughalt.de/wp-content/uploads/2011/10/DB11321.jpg

What towns would you suggest to base yourself for 3 days